Theorem.If $\displaystyle \sum_{0}^{n} a_{n} $ converges then $\displaystyle f(z) = \sum_{0}^{\infty} a_{n}z^{n} $ tends to $\displaystyle f(1) $ as $\displaystyle z $ approaches $\displaystyle 1 $ in such a way that $\displaystyle |1-z|/(1-|z|) $ remains bounded.

Here was assuming that $\displaystyle R=1 $ and convergence takes place at $\displaystyle z = 1 $.

But we let $\displaystyle R = k $ and $\displaystyle z = k $ for example, right?